Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

Satoshi Masaki; Jun Ichi Segata

doi:10.1090/tran/7262

Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

Satoshi Masaki, Jun Ichi Segata

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂_t ² − Δ is d’Alembertian. For a given asymptotic profile u_ap, we construct a solution u to (NLKG) which converges to u_ap as t → ∞. Here the asymptotic profile u_ap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

Original language	English
Pages (from-to)	8155-8170
Number of pages	16
Journal	Transactions of the American Mathematical Society
Volume	370
Issue number	11
DOIs	https://doi.org/10.1090/tran/7262
Publication status	Published - 2018
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)
Applied Mathematics

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10.1090/tran/7262

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title = "Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions",

abstract = "In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂t 2 − Δ is d{\textquoteright}Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.",

author = "Satoshi Masaki and Segata, {Jun Ichi}",

note = "Funding Information: Received by the editors November 25, 2016, and, in revised form, April 17, 2017. 2010 Mathematics Subject Classification. Primary 35L71; Secondary 35B40, 81Q05. Key words and phrases. Scattering problem. The first author was partially supported by the Sumitomo Foundation, Basic Science Research Projects No. 161145. The second author was partially supported by JSPS, Grant-in-Aid for Young Scientists (A) 25707004. Publisher Copyright: {\textcopyright} 2018 American Mathematical Society.",

year = "2018",

doi = "10.1090/tran/7262",

language = "English",

volume = "370",

pages = "8155--8170",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "11",

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T1 - Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

AU - Masaki, Satoshi

AU - Segata, Jun Ichi

N1 - Funding Information: Received by the editors November 25, 2016, and, in revised form, April 17, 2017. 2010 Mathematics Subject Classification. Primary 35L71; Secondary 35B40, 81Q05. Key words and phrases. Scattering problem. The first author was partially supported by the Sumitomo Foundation, Basic Science Research Projects No. 161145. The second author was partially supported by JSPS, Grant-in-Aid for Young Scientists (A) 25707004. Publisher Copyright: © 2018 American Mathematical Society.

PY - 2018

Y1 - 2018

N2 - In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂t 2 − Δ is d’Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

AB - In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂t 2 − Δ is d’Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -

Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

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